On at systems behaviors and observable image representations

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1 Available online at Systems & Control Letters 51 (2004) On at systems behaviors and observable image representations H.L. Trentelman Institute for Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, Netherlands Received 17 February 2003; received in revised form 27 March2003; accepted 30 June 2003 Abstract In this short paper we propose a denition of atness for systems not necessarily given in input/state/output representation. A at system is a system for which there exists a mapping such that the manifest system behavior is equal to the image of this mapping, and such that the latent variable appearing in this image representation can be written as a function of the manifest variable and its derivatives up to some order. For linear dierential systems, atness is equivalent to controllability. We will generalize the main theorem of Levine and Nguyen (Systems Control Lett. 48 (2003) 69) to general linear dierential systems. c 2003 Elsevier B.V. All rights reserved. Keywords: Flat systems; Behaviors; Observable image representations; Controllability 1. Introduction In recent papers [1 3], dierentially at systems have been studied, and their relevance in control problems has been outlined. In particular, in [3], linear at systems represented in terms of polynomial matrices have been considered, and for a particular class of such systems at outputs have been characterized in terms of their so-called dening matrices. The aim of this short paper is to explain how the notion of atness ts naturally into a behavioral perspective to systems. In fact, at systems are systems whose system behavior admit an image representation in which the latent variable is observable from the manifest system variable. In this short paper we will generalize the main theorem of [3] to general linear dierential systems. Tel.: ; fax: address: h.l.trentelman@math.rug.nl (H.L. Trentelman). 2. Flat systems In this section we will briey review the notion of dierentially at system as was introduced in [1 3]. In the sequel, C (R; R n ) will denote the space of innitely often dierentiable functions from R to R n. Let f : R n R m R n be a given function, and consider the system d x(t)=f(x(t);u(t)) with x(t) R n being the state and u(t) R m the input. This system is called dierentially at, or just at, if there exists a set of independent variables (to be called a at output of the system) such that both the system variables x and u are functions of this at output and a nite number of its successive derivatives. To be precise, if there exist nonnegative integers p and q, and functions : R (p+1)m R n, : R (p+1)m R m, and h : R n R (q+1)m R m such that the following two /$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi: /s (03)

2 52 H.L. Trentelman / Systems & Control Letters 51 (2004) conditions hold: 1. (x; u) C (R; R n R m ) satises the dierential equation (d=)x=f(x; u) if and only if there exists a function y C (R; R m ) suchthat x = (y; y (1) ;y (2) ;:::;y (p) ); u = (y; y (1) ;y (2) ;:::;y (p) ); 2. for all y C (R; R m ) we have: x = (y; y (1) ; y (2) ;:::;y (p) ) and u = (y; y (1) ;y (2) ;:::;y (p) ) implies y = h(x; u; u (1) ;:::;u (q) ). In eect, atness of the system (d=)x=f(x; u) means that the space of solutions (x; u) of the dierential equation can be represented as the image of some mapping C (R; R m ) C (R; R n ) C (R; R m ), ( (y; y (1) ;y (2) ;:::;y (p) ) ( ) ) x y = : (y; y (1) ;y (2) ;:::;y (p) ) u In addition, to any given (x; u) corresponds a unique y, which can be obtained as y = h(x; u; u (1) ;:::;u (q) ). In [3], any such y is called a at output of the system. Note that, by denition, the number of at output components is equal to m, the number of input components of the system. It was shown in [2] that the linear state-space system (d=)x = Ax + Bu (with x(t) R n and u(t) R m )is at if and only if the pair (A; B) is controllable. In [3] linear systems of the form A(d=)x = Bu were considered, with A() a given real polynomial matrix, B a given real constant matrix, and x(t) R n, u(t) R m. Suchsystem was called linearly at if it is at, and if the functions, and h can be chosen to be linear. Hence, the system A(d=)x =Bu is linearly at if and only if there exist real polynomial matrices P(), Q() and L(), respectively, of dimensions n m, m m and m (n+m), suchthat (x; u) C (R; R n R m ) satises the dierential equation A(d=)x = Bu if and only if there exists a function y C (R; R m ) suchthat ( ) ( ) d x P = ( ) u d y; Q and suchthat for all y C (R; R m ) we have: x = P(d=)y and u = Q(d=)y implies y = L(d=)( x u ). 3. Flat system behaviors In this section we will extend the notion of atness to more general systems, not necessarily in input/state/output representation. Let q, r and s be given nonnegative integers, let f : R (s+1)q R r be a given function, and consider the higher order nonlinear dierential equation in the unknown w C (R; R q ): f(w(t);w (1) (t);:::;w (s) (t))=0: (3.1) The subset B C (R; R q ) of all solutions to the dierential equation (3.1) is called the behavior of the dierential system represented by the dierential equation (3.1), w is called the manifest variable of the system. The system behavior B will be called at if there exist nonnegative integers k, l, p, and functions g : R (k+1)l R q and h : R (p+1)q R l such that the following two conditions hold: 1. B = {w C (R; R q ) there exists C (R; R l ) suchthat w = g( ; (1 );:::; (k) )}, 2. for all C (R; R l ) we have: w = g( ; (1 );:::; (k) ) implies = h(w; w (1) ;:::;w (p) ). In other words, the system behavior B represented by (3.1) is called at if B can be represented as the image of some map C (R; R l ) C (R; R q ), g( ; (1 );:::; (k) ), with the property that can be recovered from the given manifest variable trajectory w by = h(w; w (1 );:::;w (p) ). In the terminology of the behavioral approach, the variable in the above is called a latent variable, and the representation w = g( ; (1 );:::; (k) ) is called an image representation of B. The image representation is observable in the sense that the latent variable trajectory is uniquely determined by the manifest variable trajectory w through = h(w; w (1 );:::;w (p) ). Note that, in contrast to the denition of atness in [3], in our denition the number of components of the at latent variable is not required to be equal to the number of inputs of the system. Clearly, by taking w =(x; u), we see that atness of the system (d=)x = f(x; u) in the sense [3] implies atness in the sense of our denition. We will now turn attention to linear dierential systems. A linear dierential system is a system in

3 H.L. Trentelman / Systems & Control Letters 51 (2004) which the function f is linear. In that case the representing dierential equation is of the form R 0 w(t) + R 1 w (1) (t) + + R s w (s) (t) = 0 for given real r q matrices R i. After introducing a real r q polynomial matrix R() :=R 0 + R R s s, this dierential equation can be written as R ( d ) w =0: (3.2) The subspace B C (R; R q ) of all solutions to (3.2) is called the behavior of the linear dierential system represented by R(d=)w = 0. The system behavior B will be called linearly at if there exists a nonnegative integer l and real polynomial matrices M() and L() of sizes q l and l q, respectively, such that the following two conditions hold: (1) B={w C (R; R q ) there exists C (R; R l ) suchthat w = M(d=) }, (2) for all C (R; R l ) we have: w = M(d=) implies = L(d=)w. If (1) and (2) above hold, then = L(d=)w is called a linear at latent variable for B. Clearly, if the linear dierential system A(d=)x = Bu studied in [3] is linearly at in the sense of [3] then it is at in the sense of our denition. 4. Controllable behaviors and image representations We will now quickly review the notions of controllability and observable image representations in a behavioral framework. Let R() be a real r q polynomial matrix and consider the linear dierential system behavior B represented by R(d=)w = 0.B is called controllable if for any two trajectories w 1 and w 2 in B there exists T 0 and a trajectory w in B suchthat w(t)=w 1 (t) for t 0 and w(t)=w 2 (t) for t T. It is well known, see for example [5], that B is controllable if and only if the following condition on the polynomial matrix R() holds: rank(r()) = rank(r) for all C: In other words, if for any complex number, the rank of the complex matrix R() is equal to the rank of the polynomial matrix R(). Whereas a linear dierential system behavior is dened as the space of solutions B of a dierential equation of the form R(d=) = 0, it can have other representations as well. One of these is the image representation. Let M() be a real polynomial matrix with q rows and, say, l columns. If B = {w C (R; R q ) there exists C (R; R l ) suchthat w = M(d=) }; then we call w = M(d=) an image representation of the system behavior B. The image representation is called an observable image representation if from w = M(d=) 1 = M(d=) 2 it follows that 1 = 2, in other words, if any w in the behavior is the image of exactly one latent variable trajectory. Not all linear dierential systems behaviors admit an image representation. In fact, the linear dierential system behavior B admits an image representation if and only if it is controllable. In that case it also admits an observable image representation. Furthermore, observability of an image representation w = M(d=), with taking its values in R l, can be tested in terms of the q l polynomial matrix M(): the image representation w = M(d=) is observable if and only if rank(m()) = l for all C: Another result that we will use is the following. Suppose M 1 () and M 2 () are q l polynomial matrices. Then w = M 1 (d=) and w = M 2 (d=) are observable image representations of the same linear dierential behavior B if and only if there exists a unimodular l l polynomial matrix W () suchthat M 1 ()=M 2 ()W (). A detailed discussion of these standard result in the behavioral theory to linear systems can be found in [5], or in [7]. We now briey recall the concept of input cardinality of a linear dierential system. Given a linear dierential system behavior B withmanifest variable w, the condition w B leaves some of the components of w free, in the sense that these components can be chosen to be arbitrary functions in C (R; R). The number of these free components is equal to the number of inputs of the behavior B, and is denoted by m(b), called the input cardinality of B. This number can be computed in terms of the polynomial matrices R() in any kernel representation R(d=)w = 0,

4 54 H.L. Trentelman / Systems & Control Letters 51 (2004) and in terms of the polynomial matrix M() inany image representation w = M(d=) of B. In fact, m(b)=q rank(r) = rank(m): In particular, the input cardinality of B is equal to the number of latent variable components in any observable image representation of B. Finally, we state the following useful well-known result. Suppose B 1 and B 2 are controllable linear differential systems suchthat B 1 B 2. Then m(b 1 )= m(b 2 ) (equality of the input cardinalities) implies that B 1 = B 2, see [7]. 5. Flat system behaviors and observable image representations In this section we will formulate and prove our main result, which generalizes [3, Theorem 1]. Theorem 5.1. Let R() be a real r q polynomial matrix, and let B be the linear dierential system represented by R(d=)w =0. Then the following statements are equivalent: 1. B is a linearly at system, 2. B is controllable, 3. B admits an image representation, 4. B admits an observable image representation. In the rest of this theorem statement, assume that any these equivalent conditions on B hold. Then the number of components of any linear at latent variable for B is equal to m(b) =q rank(r), the input cardinality of B. Furthermore, polynomial matrices M() and L() dening an (observable) image representation w = M(d=) together with a linear at latent variable = L(d=)w for B can be obtained from the polynomial matrix R() as follows: put l := q rank(r), for M() take any q l polynomial matrix such that R()M()=0and such that rank(m()) = l for all C, for L() take any polynomial left inverse of M(), i.e., any l q polynomial matrix such that L()M()=I l l. Finally, for any q l polynomial matrix M () we have: w = M (d=) is an observable image representation of B if and only if there exist a l l unimodular polynomial matrix W () such that M ()= M()W (). In that case, =L (d=)w, with L () := W 1 ()L(), is the at latent variable corresponding to the image representation w = M (d=). Proof. The equivalence of statements (2) (4) is standard in the behavioral approach, see for example [5]. The implication (1) (3) follows from the denition of linearly at system. We prove the implication (4) (1). Let w = M(d=) be an observable image representation of B. Then M() has full column rank for all C, so th e Smithform of the polynomial matrix M() equals (I 0) T. Consequently M() has a polynomial left inverse, say L(). Clearly w = M(d=) then implies L(d=)w = L(d=)M(d=) =. The statement about the number of components of any at latent variable follows from the fact that the number of latent variables in any observable image representation of B is equal to m(b). For completeness, we also prove the remaining statements on the computation of M() and L(). Assume M() is a q l polynomial matrix suchthat R()M() = 0. Dene B := {w C (R; R q ) there exists C (R; R l ) suchthat w = M(d=) }: Then it is clear that B B. Since M() has full column rank l for all C, M() also has full column rank l as a polynomial matrix. Hence we have m(b )=l. Since this is equal to m(b), and since both B and B are controllable, we in fact have B = B. We conclude that w = M(d=) is indeed an image representation of B. Since rank(m()) = l for all, it is observable. Finally, as already explained above, by taking a polynomial left inverse L() of M() we get a linear at latent variable = L(d=)w for B. The remaining statements follow from the fact that for any two q l polynomial matrices M 1 and M 2, dening observable image representations of one and the same behavior B, there exists a unimodular matrix W suchthat M 1 = M 2 W.

5 H.L. Trentelman / Systems & Control Letters 51 (2004) The actual computation of one suitable pair of polynomial matrices M() and L(), starting from the representing polynomial matrix R() can be made more concrete as follows. First note that, by controllability, rank(r()) = rank(r)= : r for all C. Hence the nonzero polynomials in the Smith form of R() are all equal to 1, so there exist unimodular matrices U() and V () suchthat ( ) Ir r 0 U()R()V ()= : 0 0 Dene then ( ) 0 M() :=V() ; L() :=(0 I (q r) (q r) I (q r) (q r) ) V 1 (): Remark 5.2. The question arises as to what extend the result of Theorem 5.1 can be extended to more general classes of systems. In general, nonlinear systems, even controllable ones, have an image representation only in exceptional cases. Another class of systems of interest is the class of ND-systems, i.e. systems represented by a nite number of linear, constant coecient partial dierential equations. It has been shown in [4] for ND-systems that the existence of an image representation is equivalent to controllability. However, only in exceptional cases a controllable ND-system admits an observable image representation. In [6], for ND-systems the property of admitting an observable image representation has been shown to be equivalent to the property of rectiability, which, in turn, has been shown to be equivalent to strong controllability. Acknowledgements The author wishes to thank Dr. O. Iftime for drawing his attention to the subject of at systems. References [1] M. Fliess, J. Levine, Ph. Martin, P. Rouchon, Flatness and defect of nonlinear systems: introductory theory and applications, Internat. J. Control 61 (1995) [2] M. Fliess, J. Levine, Ph. Martin, P. Rouchon, A Lie Backlund approachto equivalence and atness of nonlinear systems, IEEE Trans. Automat. Control 44 (5) (1999) [3] J. Levine, D.V. Nguyen, Flat output characterization for linear systems using polynomial matrices, Systems Control Lett. 48 (2003) [4] H.K. Pillai, S. Shankar, A behavioral approach to control of distributed systems, SIAM J. Control Optim. 37 (1998) [5] J.W. Polderman, J.C. Willems, Introduction to Mathematical System Theory: A Behavioral Approach, Springer, Berlin, [6] P. Rocha, J. Wood, A new perspective on controllability properties for dynamical systems, J. Appl. Math. Comput. Sci. 7 4 (1997) [7] J.C. Willems, H.L. Trentelman, Synthesis of dissipative systems using quadratic dierential forms Part I, IEEE Trans. Automat. Control 47 (1) (2002)